We have the following setting
We have a 2-dimensional Brownian motion $(X,Y)$, and we define the process $M_t$ as $$M_t=e^{X_t}\cos(Y_t)$$
The problem is to show that the process $M_t$ is a local martingale, and that we have that $$\langle M\rangle_t=\int^t_0e^{X_s}ds$$
I know that $M_t$ would be a local martingale if it is a continuous, adapted process and if there exists a sequence of stopping times $(\tau_n)_{n\geq0}$, such that $\tau_n\uparrow\infty$ almost surely, and that $(M_{\tau_n\wedge t}-M_0)_{t\geq0}$ is a uniformly integrable martingale for all $n$.
However, I am not aware of any straight forward way to proving that something is a local martingale, and I do not see an easy approach to proving it directly from the definition. Are there ''standard'' routes to follow when trying to prove that a process is/isn't a local martingale?
Moreover, when trying to show that the quadratic variation of $M$ can be written in such a way, I cannot figure out why the cosine term would drop out.
Any help is appreciated!